Linear difference equations with transition points

Abstract:

Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation \begin{equation} y_{n+1}(x)-(A_nx+B_n)y_n(x)+y_{n-1}(x)=0, \nonumber \end{equation} where $A_n$ and $B_n$ have power series expansions of the form \begin{equation} A_n\sim \sum ^\infty _{s=0}\frac {\alpha _s}{n^s}, \qquad \qquad B_n\sim \sum ^\infty _{s=0}\frac {\beta _s}{n^s}\nonumber \end{equation} with $\alpha _0\ne 0$. Our results hold uniformly for $x$ in an infinite interval containing the transition point $x_+$ given by $\alpha _0 x_++\beta _0=2$. As an illustration, we present an asymptotic expansion for the monic polynomials $\pi _n(x)$ which are orthogonal with respect to the modified Jacobi weight $w(x)=(1-x)^\alpha (1+x)^\beta h(x)$, $x\in (-1,1)$, where $\alpha$, $\beta >-1$ and $h$ is real analytic and strictly positive on $[-1, 1]$.